Composition of functions is a method of combining two or more functions into one while an inverse function undoes the operation of composition. Composition is done by using the output of one function as the input of another. Alternately, composite functions can be broken down into simpler functions. The operation of composition is associative, but not commutative. The function $f(x)=x$ is the identity function for composition. If the inverse of a function is defined, the composition of a function and its inverse is the identity function.

### At A Glance

- The composition of functions is defined by using the output of one function as the input of another.
- To evaluate the composition of two functions $f(g(x))$ for a value of $x$, first find $g(x)$ and then substitute the value in $f$, or find a general rule for $f(g(x))$ and then evaluate for the value of $x$.
- To determine the rule for a composite function $f(g(x))$, substitute $g(x)$ into $f(x)$ and simplify. The domain of $f(g(x))$ is the set of values in the domain of $g$ for which $g(x)$ is in the domain of $f$.
- Composite functions can be decomposed into two or more functions using methods such as the properties of operations, the properties of exponents, and the word form of the function.
- Since composition of functions is associative, the way in which three or more functions are grouped in a composite function does not change the result.
- Composition is not commutative, so composing functions in a different order does not necessarily yield the same result.
- The identity function for the operation of composition is $f(x) = x$.
- A function is one-to-one if every output value has a unique input value. The horizontal line test and algebraic methods are used to determine whether a function is one-to-one.
- Two functions $f$ and $g$ are inverse functions if $f(g(x)) = g(f(x)) = x$. To determine the rule for an inverse function, write as $y = f(x)$, reverse $x$ and $y$, and then solve for $y$. The domain of $f$ is the range of $g$, and vice versa.
- The graphs of inverse functions are reflections of each other across the line $y = x$.
- If the inverse of a function is not a function, the domain of the function can be restricted to values that produce a function for the inverse.